Cylindrical Coordinates — Question 8

Question 8

A moving point has cylindrical coordinates $r(t),\theta(t),z(t)$. Derive its rectangular velocity and prove that its speed satisfies $$\|\vec v\|^2=\dot r^{\,2}+r^2\dot\theta^{\,2}+\dot z^{\,2}.$$

Question 8 – Solution

Strategy Differentiate $x=r\cos\theta$ and $y=r\sin\theta$, then simplify the sum of squares.

Velocity $$\dot x=\dot r\cos\theta-r\dot\theta\sin\theta,\quad \dot y=\dot r\sin\theta+r\dot\theta\cos\theta,\quad \dot z=\dot z.$$

Speed identity Expanding $\dot x^2+\dot y^2$ makes the cross terms cancel, while $\sin^2\theta+\cos^2\theta=1$. Therefore $$\boxed{\|\vec v\|^2=\dot x^2+\dot y^2+\dot z^2 =\dot r^{\,2}+r^2\dot\theta^{\,2}+\dot z^{\,2}}.$$ The radial, angular, and vertical velocity directions are mutually perpendicular.